Linear differential systems with small coefficients: various types of solvability and their verification

We study the problem of solvability of linear differential systems with small coefficients in the Liouvillian sense (or, by generalized quadratures). For a general system, this problem is equivalent to that of solvability of the Lie algebra of the differential Galois group of the system. However, dependence of this Lie algebra on the system coefficients remains unknown. We show that for the particular class of systems with non-resonant irregular singular points that have {\it sufficiently small} coefficient matrices, the problem is reduced to that of solvability of the explicit Lie algebra generated by the coefficient matrices. This extends the corresponding Ilyashenko--Khovanskii theorem obtained for linear differential systems with Fuchsian singular points. We also give some examples illustrating the practical verification of the presented criteria of solvability by using general procedures implemented in Maple.


Introduction
Solvability of linear differential equations and systems in finite terms is a classical question of differential Galois theory. It begins with the problem of solvability by generalized quadratures (or, in the Liouvillian sense, which means the representability of all solutions of an equation in terms of elementary or algebraic functions and their integrals, speaking informally) in the 1830's in the works of Liouville on second order equations. Generalized in 1910 by Mordukhai-Boltovskii for nth order equations, this was independently developed in a quite different way by Picard and Vessiot who connected to an equation (system) a group, turned out to be a linear algebraic group, called the differential Galois group. They showed that solvability of the equation (system) in the Liouvillian sense depends entirely on properties of this group or its Lie algebra. Namely, solvability holds if and only if the identity component of the differential Galois group is solvable (which is equivalent to the solvability of the Lie algebra of the group). Later Kolchin completed this theory by considering other, more particular, types of solvability and their dependence on properties of the differential Galois group.
General methods for computing (in theory, at least) the differential Galois group have been proposed by several authors [5], [9], [6], [17] in the last past decades. However, those methods have an extremely high computational complexity and are therefore not practical for nontrivial problems. Moreover it is not clear how the differential Galois group of a given system depends on the coefficients of the latter. More recently an algorithm for computing the Lie algebra of the differential Galois group has been proposed in [3]. This algorithm has a rather reasonable complexity, compared to the previous methods for computing the differential Galois group, but as the former it does not allow to clarify the relationship between the coefficients of the input system and its differential Lie algebra.
Though the main difficulty is that the differential Galois group (and its Lie algebra) of a specific equation depends on its coefficients very implicitly, in some cases it is possible to leave aside these implicit objects (which are hard to compute) and obtain an answer to the question of solvability in terms of objects that are determined by the coefficients of the equation explicitly. Here are some examples: • the list of hypergeometric equations solvable in the Liouvillian sense is completely known (Scwharz-Kimura's list consisting of the fifteen families [13], see also [22,Ch.12,§1]); • the Bessel equation is solvable in the Liouvillian sense if and only if its parameter is a half-integer (see [16, §2.8] or [22,Ch.11,§1]); • the equation y ′′ = (az 2 + bz + c)y, a = 0, is solvable in the Liouvillian sense if and only if (4ac − b 2 )/4a 3/2 is an odd integer (see [15]); • more generally, an equation y ′′ = P (z)y with a polynomial P , is not solvable in the Liouvillian sense if deg P is odd (see [10]) whereas in the (2n + 1)-dimensional space of such equations with deg P = 2n, equations solvable in the Liouvillian sense form a union of countable number of algebraic varieties of dimension n + 1 each (see [18], [21]).
A rather unexpected and maybe less known subclass of linear differential systems whose various types of solvability can be checked in terms of explicit input data, is formed by systems with sufficiently small matrix coefficients. The first result of this kind concerns Fuchsian systems. It was obtained by Ilyashenko, Khovanskii and published in 1974 in Russian, and is revised in [11], [12,Ch.6,§2.3]. This claims that for Fuchsian (p × p)- with fixed singular points a 1 , . . . , a n ∈ C (and maybe ∞, if n i=1 B i = 0) there exists an ε = ε(p, n) > 0 such that a criterion of solvability in the Liouvillian sense for a Fuchsian system with B i < ε takes the following form: the system is solvable if and only if all the matrices B i can be simultaneously reduced to a triangular form. Using Kolchin's results, Ilyashenko and Khovanskii also obtained criteria for other types of solvability of a Fuchsian system with small residue matrices, in terms of these matrices. Though, all these criteria still remain not quite explicit in the sense that the ε is expressed implicitely. This situation was refined in [19] where it was proved that for applying the above criterion of solvability it is sufficient for the eigenvalues of the matrices B i , called the exponents of the system, to be small enough rather than for the matrices themselves (this refinement had been conjectured earlier by Andrey Bolibrukh, see remark in section 6.2.3 of [11]). Moreover, an explicit bound for the exponents was given. Then similar criteria of solvability were obtained in [8] for a (non-resonant) irregular differential system with small formal exponents.
In the next section we recall these criteria for various types of solvability of a linear differential system with small (formal) exponents (Theorems 1, 1', 2, 3), previously giving necessary definitions. Then, in Section 3, we prove that the formal exponents at a nonresonant irregular singular point z = a of a system (1) are small, if the coefficient matrices B (1) , . . . , B (r) of the principal part of B(z) at a are small enough, while the leading term B (0) belongs to some compact subset of Mat(p, C). This allows us to formulate criteria of solvability inside an open set of non-resonant irregular differential systems (Theorem 4), looking at the problem theoretically, like Ilyashenko and Khovanskii did in the Fuchsian case.
From a practical point of view, as soon as we have a system whose (formal) exponents satisfy some numerical restrictions, which can be checked algorithmically, the question on solvability of the implicit Lie algebra of the differential Galois group of the system is reduced to the question on solvability of the explicit Lie algebra generated by the system coefficient matrices. This practical part of the problem is studied in Section 4, where the corresponding algorithms and examples are provided.

Various types of solvability
We consider a linear differential (p × p)-system defined on the whole Riemann sphere C, with the meromorphic coefficient matrix B (whose entries are thus rational functions). One says that a solution y of the system (2) is Liouvillian if there is a tower of elementary extensions of the field C(z) of rational functions such that all components of y belong to F m . Here each F i+1 = F i x i is a field extension of F i by an element x i , which is either: -an integral of some element in F i , or -an exponential of integral of some element in F i , or -algebraic over F i .
The system is said to be solvable in the Liouvillian sense (or, by generalized quadratures), if all its solutions are Liouvillian. There are other types of solvability studied by Kolchin [14] from the point of view of differential Galois theory, which are defined in analogy to solvability by generalized quadratures, and we leave formal definitions to the reader. These are: 1. solvability by integrals and algebraic functions; 2. solvability by integrals; 3. solvability by exponentials of integrals and algebraic functions; 4. solvability by algebraic functions.
We leave aside the differential Galois group of the system and the description of the above types of solvability in terms of this group, since in the case of systems having small (formal) exponents we will deal with, this description is provided in terms of the coefficient matrix. Let us recall some basic theory. Assume that the system (2) has singular points a 1 , . . . , a n ∈ C of Poincaré rank r 1 , . . . , r n respectively and, probably, a singular point at infinity. This means that the coefficient matrix B has the form The polynomial part is absent and n i=1 B Further we assume, for the simplicity of exposition, that the point ∞ is indeed non-singular, though we will consider some example with a singular point at infinity (Example 1 below). We also restrict ourselves to the generic case, when each singular point a i is either Fuchsian or irregular non-resonant. In the first case the Poincaré rank r i equals zero, while in the second case it is positive and the eigenvalues of the leading term B (0) i are pairwise distinct. Near a Fuchsian singular point z = a i , the system possesses a fundamental matrix Y of the form where the matrix U is holomorphically invertible at a i (that is, det U(a i ) = 0), A is a diagonal integer matrix, E is a triangular matrix. The eigenvalues λ 1 i , . . . , λ p i of the triangular matrix A + E are called the exponents of the system at a Fuchsian singular point a i and they coincide with the eigenvalues of the residue matrix B Near a non-resonant irregular singular point z = a i , the system possesses a formal fundamental matrix Y of the form F is a matrix formal Taylor series in (z − a i ), with det F (a i ) = 0, and Q is a polynomial. The numbers λ 1 i , . . . , λ p i are called the formal exponents of the system at the non-resonant irregular singular point a i , and the algorithm of their calculation will be recalled in the next section. Now we give some theorems which follow from [8], [7] and which we will be based on further.
Theorem 1. Let the exponents of the system (2), (3) at all singular points satisfy the condition Re λ j i > −1/n(p − 1), i = 1, . . . , n, j = 1, . . . , p, furthermore, for every Fuchsian singular point a i each difference λ j i −λ k i ∈ Q\Z. Then the system is solvable by generalized quadratures if and only if there exists a constant matrix C ∈ GL(p, C) such that all the matrices CB According to the behaviour of solutions of a linear differential system near its irregular singular point, the system (2) with at least one irregular singular point is not solvable by integrals and algebraic functions. For solvability by exponentials of integrals and algebraic functions, the following criterion holds.  Example 1. Let us give an example which illustrates that to expect the equivalence of solvability in the Liouvillian sense to the triangularizability of a system via a constant gauge transformation, one indeed need to put some restrictions on the (formal) system exponents.
Consider the equation Written in the form of a (2 × 2)-system with respect to the vector of unknowns y = (u, du/dz) ⊤ , this becomes dy dz = 0 1 z 2 + c 0 y, or, in the variable t = 1/z, This is a system with the unique singular point t = 0 of Poincaré rank 3 which is resonant though, since the leading term of the coefficient matrix is nilpotent. Thus we make the transformationỹ = t diag(0,1) y, under which the coefficient matrix B(t) of the system is changed as follows: The transformed system has two singular points  Here we prove the following statement generalizing the Ilyashenko-Khovanskii criterion of solvability to the non-resonant irregular case. This theorem will clearly follow from Theorem 5 below and Theorem 1.
Proof. To prove the statement of the theorem, let us first recall the procedure of a formal transformation of the system (1) to a diagonal form, the Splitting lemma (from [20, §11], and we assume that a = 0 for simplicity).
Under a transformation y = T (z)ỹ, the system is changed as follows: and a new coefficient matrix A has the Laurent expansion One may always assume that T (z) = I + T (1) z + . . . and B (0) = A (0) = diag(α 1 , . . . , α p ). Then gathering the coefficients at each power z k from the relation (the last summand equals zero for k r). There are two sets of unknowns in this system of matrix equations: ij ). Requiring all the A (k) 's to be diagonal and assuming A (0) , . . . , A (k−1) and T (0) , . . . , T (k−1) to be already found, one first obtains A where H (k) = k−1 l=1 (T (l) A (k−l) − B (k−l) T (l) ) + (k − r)T (k−r) , and then Thus one can see that Therefore, jj . This implies the estimate (we will use, for example, the matrix 1-norm · 1 here). Now we are ready to prove the smallness of the formal exponents λ j 's in the case of small coefficients B (1) , . . . , B (r) . For this, we should prove the smallness of H (1) , . . . , H (r) .
There are two simple corollaries of Theorem 4.
Corollary 1 (for systems with singular points of Poincaré rank 1). There exists δ = δ(p, n) > 0 such that in a set systems solvable by generalized quadratures are those and only those whose matrices B i , i = 1, . . . , n, are triangular (in some common basis). Proof. If all the Poincaré ranks r i = 1, then |λ j i | B (1) i and thus one has no necessity to restrict the leading terms B

Practical remarks and examples
We conclude by noting that the results presented in this paper can be turned into an algorithm which allows to check the solvability of a given concrete system. The main steps can be summarized as follows. 2. Check the required smallness of the formal exponents of the system under consideration.
3. The simultaneous triangularizability of a set of matrices is equivalent to the solvability of the Lie algebra generated by them. This can be checked using the package LieAlgebras of Maple 4 .
In order to illustrate the various steps of our algorithm we give two examples of computations performed by using our implementation in Maple.
Example 2. We consider a (3 × 3)-system (2) with the coefficient matrix where the matrices M i are given by and a, b and c are parameters. It has three singular points a 1 = 0, a 2 = 1, a 3 = −1 of Poincaré rank r 1 = 2, r 2 = 1, r 3 = 1, respectively. The point z = ∞ is non-singular, since Our implementation allows to check that the three singularities are non-resonant and gives the formal exponents for each of them.

> splitlemma(B, z=-1, 1, T);
We get the equivalent matrix  The formal exponents at z = a 3 are then λ 1 3 = −a, λ 2 3 = −c, λ 3 3 = −b. Note that in our case (p = 3, n = 3) the condition (4) of Theorem 1 is as follows: This is equivalent here to the condition |Re λ| < 1/6, for λ ∈ {a, b, c}.  This tells us that the Lie algebra L is sovable (apparently for all a, b, c) but it does not give a transformation P that simultaneously triangularizes the matrices M i . Such a transformation can be directly obtained using the implementation from [2].
In conclusion, the system in this example is solvable in the Liouvillian sense without any restriction on the parameters a, b, c. On the other hand, the matrices M i , i = 1, . . . , 6, are never simultaneously diagonalizable since, for example, M 1 , M 2 do not commute: Hence, due to Theorem 2 we can assert that the system is not solvable by exponentials of integrals and algebraic functions whenever |Re λ| < 1/6, for λ ∈ {a, b, c}.
Example 3. We consider a (3 × 3)-system (2) with the coefficient matrix where the matrices M i are given by This should be interpreted as that the Lie algebra L is not solvable (and hence our three matrices M i are not simultaneously triangularizable) for generic values of a, b. Therefore, the system is not solvable by generalized quadratures for generic values of a, b.
Our implementation [2] gives the same answer: This means that for a = 0, b = −5, the system is solvable by generalized quadratures and, on the other hand, is not solvable by exponentials of integrals and algebraic functions (since the matrices N i are not simultaneously diagonalizable: [N 1 , N 2 ] = 0).